Tight Approximation Ratio of a General Greedy Splitting Algorithm for the Minimum k-Way Cut Problem

Tigh t Appro ximation Ratio of a General Greedy Splitting Algorithm for the Minim um k -W a y Cut Problem Mingyu Xiao Leizhen Cai Andrew C. Y ao Departmen t of Computer Science and Engineering The Chinese Univ ersity of Hong Kong Hong Kong SAR, CHINA Email: myx iao(lcai )@cse.cuhk.edu.hk, andre wcyao@t singhua.edu.cn Abstract F o r an edge-w eighted connected undirected graph, the minimum k -way cut pr oblem is to find a subset of edges o f minim um total weight whose remov al separates the gra ph in to k co nnec ted compo nen ts. The pr oblem is NP-hard when k is pa rt o f the input and W[1]-hard when k is tak en a s a parameter. A simple alg orithm for approximating a minimum k -wa y cut is to iter- atively incr ease the n umber of comp onent s of the graph b y h − 1, wher e 2 ≤ h ≤ k , until the gr a ph has k co mponents. The approximation ratio of this algor ithm is known for h ≤ 3 but is op en for h ≥ 4. In this pa per, we consider a general alg o rithm that iter ativ ely increa s es the n um b er of co mponents of the g raph by h i − 1, where h 1 ≤ h 2 ≤ · · · ≤ h q and P q i =1 ( h i − 1) = k − 1. W e prove that the approximation ratio of this general algor ithm is 2 − ( P q i =1  h i 2  ) /  k 2  , which is tight. Our r esult implies that the approximation ratio of the simple algor ithm is 2 − h/k + O ( h 2 /k 2 ) in genera l and 2 − h/k if k − 1 is a multiple of h − 1. Key w ords approximation algo rithm, k -wa y c ut, k -wa y split. 1 Appro ximation r atio for k -w a y cuts 2 1 In tro duction Let G = ( V , E ; w ) a connected un directed graph with n vertice s and m edges, where eac h edge e has a p ositiv e weigh t w ( e ), and k a p ositiv e inte ger. A k -way cut of G is a su bset of edges whose remo v al separates the graph in to k connected comp onen ts, and the minimum k -way cut problem is to fin d a k -wa y cut of minim um total w eigh t. W e note that k -wa y cuts are also referred to as k -cu ts or multi-c omp onent cuts in th e literature. The minimum k -wa y cut p r oblem is a n atural generalizati on of the classical minimum cu t problem and has been very w ell studied in the literature. Gold- sc hmidt and Ho c hbaum [1] pro ve d that the minimum k -w a y cut problem is NP- hard when k is part of the input and ga v e an O ( n (1 / 2 − o (1) ) k 2 ) al gorithm. Kamidoi et al. [2] p resen ted an O ( n 4 k/ (1 − 1 . 71 / √ k ) − 31 ) algorithm, and Xiao [3] pr esen ted an O ( n 4 k − l og k ) algorithm. These three algorithms are based on a divide-and-conquer metho d. Karger and Stein [4 ] prop osed a randomized algorithm that r uns in O ( n 2 k − 2 log 3 n ) exp ected time. Recen tly , T horup [5] ob tained an O ( n 2 k log n ) algorithm via tree pac kin g. On the other hand, Do wney et al. [6] sho wed that the problem is W[1]-hard when k is take n as a p arameter, wh ic h in dicates that it is v ery un lik ely to solve the p roblem in f ( k ) n O (1) time for an y fun ction f ( k ). W e also note th at faster algorithms are a v ailable for small k . Nagamoc hi and Ibaraki [7], and Hao and Orlin [8] solv ed th e m inim um 2-wa y cut p roblem (i.e., the m inim um cut problem) in O ( mn + n 2 log n ) a nd O ( mn log ( n 2 /m )) ti me re- sp ectiv ely . Burlet and Goldsc hmid t [9] sol ve d the min im u m 3-w a y cut pr oblem in e O ( mn 3 ) time, Nagamo c hi an d Ibaraki [10] ga ve e O ( mn k ) algorithms for k ≤ 4, and Nagamochi et al. [11] extended this resu lt for k ≤ 6. F u rthermore, Levine [12] obtained O ( mn k − 2 log 3 n ) rand omized algorithms f or k ≤ 6. In terms of appro ximation algo rithms , Saran and V azirani [13] ga v e t w o simple algorithms of appr o ximation r atio 2 − 2 / k . Naor and Rabani [14] obtained an in teger program formula tion of this prob lem with int egralit y gap 2, and Ra vi and Sinha [1 5] also d eriv ed a 2- appr o ximation algorithm via the net w ork strength metho d. A simple algorithm [13] for app ro ximating a m inim um k -w a y cut is to itera- tiv ely increase the n um b er of co mp onents of the graph b y h − 1 , where 2 ≤ h ≤ k , unt il the graph has k comp onen ts. This algorithm has an appro ximation ratio of 2 − 2 /k for h = 2 [13], and Kap o or [16] claimed th at it ac h iev es ratio 2 − α ( h, k ) for h ≥ 3, where α ( h, k ) = h/k − ( h − 2) /k 2 + O ( h/k 3 ). Unfortun ately , h is pro of for h ≥ 3 is incomplete. Later, Z hao et al. [1 7 ] established Kap o or’s cla im f or h = 3: th e ratio is 2 − 3 /k for o dd k and 2 − (3 k − 4) / ( k 2 − k ) for ev en k . How eve r, for h ≥ 4, it seems quite difficult to analyze the p erformance of this algorithm and it has b een an op en problem whether we g et a b etter appro x imation ratio with this appr oac h. In this pap er, w e consider a general alg orithm that iterativ ely increases the n umb er of comp onents o f the graph by h i − 1, where h 1 ≤ h 2 ≤ · · · ≤ h q and P q i =1 ( h i − 1) = k − 1. W e p ro v e that the appro ximation ratio of th is general Appro ximation r atio for k -w a y cuts 3 algorithm is 2 − ( P q i =1  h i 2  ) /  k 2  , whic h is tigh t. Our r esult implies that the appro ximation r atio of the sim p le algorithm is 2 − h/k + O ( h 2 /k 2 ) in general and 2 − h/k if k − 1 is a m ultiple of h − 1, wh ic h settles the op en problem mentioned earlier in the affirmativ e. The rest of the pap er is organized as follo w s. In Section 2, we formalize our general greedy splitting algorithms and p r esen t our m ain r esu lts on their appro ximation ratios. W e prov e our main results in S ectio n 3 wh ile the pro of of a pu rely analytical lemma is giv en in Section 4, and conclude with some remarks in Section 5. 2 Algorithms and main results In this s ectio n, w e form alize our greedy sp litting algorithms and present our main results on their appr o ximation ratios. W e note that Zhao et al. [18, 19] ha v e stud ied suc h algorithms for general multiw a y cut and partition problems. First w e exte nd the notion of k -wa y cuts to disconnected graphs. A k -way split of a graph is a subset of edges whose remo v al increases the n umb er of comp onen ts b y k − 1. T h erefore f or a connected graph, a k -w a y split is equiv alen t to a k -wa y cut. W e note th at the time for find in g a minimum k -w a y sp lit in a general graph is the same as finding a k -wa y cut [17]. One general appr oac h f or finding a ligh t k -w a y cut is to find minimum h i -w a y splits successiv ely for a giv en sequence ( h 1 , h 2 , · · · , h q ). Algorithm iterativ e-split ( G, k , ( h 1 , h 2 , · · · , h q )) Input: Conn ecte d graph G = ( V , E ; w ), integ er k a nd sequence ( h 1 , h 2 , · · · , h q ) of inte gers s atisfying 2 ≤ h 1 ≤ h 2 ≤ · · · ≤ h q and P q i =1 ( h i − 1) = k − 1. Output: A k -w a y cut of G . 1. F or i := 1 to q fin d a minim um h i -w a y split C i of G and let G ← G − C i . 2. Return S q i =0 C i as a k -w a y cut. A sp ecial case of the ab ov e algorithm is wh en all h i ’s in the in teger s equence, with the p ossib le exceptio n of the first one, are equal. The follo win g g ive s a precise description of this sp ecial case. Algorithm iterativ e- h -split ( G, k , h ) Input: C onnected graph G = ( V , E ; w ), int egers k a nd h . Output: A k -w a y cut of G . 1. Let p = ⌊ k − 1 h − 1 ⌋ and r = ( k − 1) mo d ( h − 1). 2. If r 6 = 0, then find a minimum ( r + 1) − w a y split C 0 of G and let G ← G − C 0 . 3. F or i := 1 to p find a m inim um h -w a y sp lit C i of G and let G ← G − C i . 4. Return S p i =0 C i as a k -w a y cut. Appro ximation r atio for k -w a y cuts 4 The ab o ve t w o algorithms run in p olynomial time if h q and h a re b oun ded b y s ome constan t, and our main results of the pap er are the f ollo wing t wo tigh t b ounds for their app ro ximation ratios. Theorem 2.1 The appr oximation r atio o f a lgorithm itera tiv e-split is 2 − P q i =1  h i 2   k 2  . Corollary 2.2 The appr oximation r atio o f a lgorithm itera tiv e- h -split i s 2 − h k + ( h − 1 − r ) r k ( k − 1) = 2 − h k + O ( h 2 k 2 ) , wher e r = ( k − 1) mo d ( h − 1) . Remark. W e note that when k − 1 is a multiple of h − 1, iterat iv e- h -split is a (2 − h/k )-appro ximation algorithm, and Corollary 2.2 for h = 3 yields a result of Zhao et al. [17]. 3 P erformance analysis In this section, we will prov e our main results on th e approximat ion ratios of our appro ximation algorithms. F or this pur p ose, w e first establish a relation b et we en the weigh t w ( C h ) of a minimum h -w a y sp lit C h and the we ight w ( C k ) of a k -w a y split C k , which will b e the main to ol in our analysis. F or con ve nience, w e allo w h = 1 (note that a minim um 1-wa y split is an empty set). F or a collec tion o f m utually disjoint su bsets V 1 , V 2 , · · · , V t ∈ V , we u se [ V 1 , V 2 , · · · , V t ] to d enote th e set of edges uv suc h that u ∈ V i and v ∈ V j for some V i 6 = V j . Lemma 3.1 L et G b e an e dge-weighte d gr aph, h ≥ 1 , and k ≥ max { 2 , h } . F or any minimum h -way sp lit C h and any k -way split C k of G , the f ol lowing holds. w ( C h ) w ( C k ) ≤ (2 − h k ) h − 1 k − 1 . (1) Pro of. First we consider the case that G is connected. In this case, C k and C h , resp ectiv ely , are k -w a y and minim um h -w a y cu ts of G , and th us C k corresp onds to a partition Π = { V 1 , V 2 , . . . , V k } of the vertex set V of G suc h that eac h V i is a comp onent of G − C k . W e can merge a ny k − ( h − 1) elemen ts in Π in to one elemen t to form a new p artition Π ′ = { V ′ 1 , V ′ 2 , . . . , V ′ h } of V . Let E (Π ′ ) = [ V ′ 1 , V ′ 2 , . . . , V ′ h ]. Then G − E (Π ′ ) has at least h comp onen ts, and therefore the w eigh t w ( E (Π ′ )) of E (Π ′ ) is at least w ( C h ). There are  k h − 1  differen t w a ys to form Π ′ , and therefore the total w eigh t W of all E (Π ′ ) is at least  k h − 1  w ( C h ). Appro ximation r atio for k -w a y cuts 5 On the other hand, w e can put an upp er b oun d on W b y relating it to the w eigh t of C k . Consider the set E ij of edges in C k b et w een V i and V j . F or a partition Π ′ , E ij ⊆ E (Π ′ ) iff V i and V j are not merged in forming Π ′ . The n umb er o f Π ′ s for whic h V i and V j are merged is  k − 2 h − 1  , implying that eac h E ij is coun ted  k h − 1  −  k − 2 h − 1  times in calculating W . Therefore W = ( k h − 1 ! − k − 2 h − 1 ! ) · w ( C k ) ≥ k h − 1 ! · w ( C h ) , whic h yields the inequalit y in the lemma. F or the case that G is d isconn ecte d, we construct a connected graph G ′ = ( V ′ , E ′ ; w ′ ) from G as follo w s: 1. Add a new v ertex v . 2. F or eac h comp onen t H of G , a dd an edge e H b et w een v and an arb itrary v ertex of H . 3. Set the weigh t of e H to ∞ . 4. Set w ′ ( e ) = w ( e ) for all other edges of G ′ . Then every k -w a y split in G is a k -w a y cut in G ′ , and ev ery min imum h -w a y split in G is a minimum h -w a y cut in G ′ . Sin ce G ′ is connected, the lemma holds for G ′ and hence f or k -wa y an d minimum h -wa y splits of G . F or conv enience, d efine for all h ≥ 1 and k ≥ max { 2 , h } , f ( k , h ) = (2 − h k ) h − 1 k − 1 . W e n ote that the b ound in Lemm a 3.1 is tigh t, whic h can b e seen b y considering a k -w a y cut and a minim um h -wa y cut of the complete graph K k . This also give s a com binatorial explanation of f ( k , h ): the ratio b et ween the num b er of edges co vered b y h − 1 vertices in K k and the num b er of edges of K k . W e also need the follo wing p rop erties of f ( k , h ) in our an alysis. F act 3.2 F unction f ( k , h ) mo notonic al ly incr e ases for h ∈ [1 , k ] and monotoni- c al ly de cr e ases for k ∈ [ h, ∞ ) . F act 3.3 F or al l a ≥ 0 , h ≥ 2 , and k ≥ a + h, f ( k − a, h )(1 − f ( k , a + 1)) ≤ f ( k , h ) . (2) Pro of. S tr aigh tforward manipulation gives f ( k − a, h )(1 − f ( k , a + 1)) = (2 − 2 a + h k ) h − 1 k − 1 ≤ f ( k , h ) . Appro ximation r atio for k -w a y cuts 6 The next inequalit y is an analytical result critical to the pro of of our main theorem. Let q ≥ 2. F or any inte gers 2 ≤ h 1 ≤ h 2 ≤ · · · ≤ h q , 0 ≤ a ≤ h 1 − 1 and k − 1 ≥ P q i =1 ( h i − 1), let D = f ( k − a, h 1 − a ) + q X i =2 f ( k − a, h i ) (3) and F = max { D , f ( k, a + 1) + (1 − f ( k , a + 1)) D } . (4) Lemma 3.4 F ≤ P q i =1 f ( k , h i ) . T o av oid distraction f rom our main discussions, w e dela y the p ro of of this purely analytical lemma to Section 4. W e are no w ready to prov e our m ain results. F or th is p urp ose, w e call a sequence (( C 1 , h 1 ) , . . . , ( C q , h q )) a nonde cr e asing q -se quenc e of minimum splits if in tegers 2 ≤ h 1 ≤ h 2 ≤ · · · ≤ h q and ea c h C i , 1 ≤ i ≤ q , is a minimum h i -w a y split of G i = G − S i − 1 j =1 G j . T o prov e Theorem 2.1, it suffices to pro v e the follo wing theorem. W e note that although the pro of is an in ductiv e one, the argument in the pr o of is su btle, and the condition h 1 ≤ h 2 ≤ · · · ≤ h q is crucial to the pr oof. Theorem 3.5 L et (( C 1 , h 1 ) , . . . , ( C q , h q )) b e a nonde cr e asing q -se q uenc e of min- imum splits of a weighte d gr aph G = ( V , E ; w ) , wher e w : E → R + , and S k a k -way split of G satisfying k − 1 ≥ P q i =1 ( h i − 1) . Then w ( q [ i =1 C i ) ≤ q X i =1 f ( k , h i ) · w ( S k ) . (5) Pro of. W e use in duction on q . F or q = 1, the theorem is established by Lemma 3.1. F or the ind uctiv e step, let q ≥ 2, C ′ 1 = C 1 ∩ S k , S k ′ = S k − C ′ 1 , and C ′′ 1 = C 1 − C ′ 1 . Then C ′ 1 is an ( a + 1)-wa y split of G for some 0 ≤ a ≤ h 1 − 1, C ′′ 1 is a minimum ( h 1 − a )-w a y sp lit of G − C ′ 1 (otherwise C 1 w ould not b e a minim um h 1 -w a y split of G ), and S k ′ is a ( k − a )-w a y split of G − C ′ 1 . It follo w s that S k ′ is a k ′ -w a y split of G − C 1 for some k ′ ≥ k − a . Note that (( C 2 , h 2 ) , . . . , ( C q , h q )) is a nond ecreasing ( q − 1)-sequence of minimum splits of G − C 1 and k ′ − 1 ≥ P q i =2 ( h i − 1). By the ind uction hyp othesis and the f act that eac h f ( k ′ , h i ) is at most f ( k − a, h i ) (F act 3.2), we h av e w ( q [ i =2 C i ) ≤ q X i =2 f ( k ′ , h i ) · w ( S k ′ ) ≤ q X i =2 f ( k − a, h i ) · w ( S k ′ ) . (6) Let W = w ( C 1 ) + P q i =2 f ( k − a, h i ) · w ( S k ′ ). Then w ( S q i =1 C i ) ≤ W by (6), and w e will establish the theorem by proving W ≤ P q i =1 f ( k , h i ) · w ( S k ). Appro ximation r atio for k -w a y cuts 7 If w ( C ′ 1 ) > f ( k , a + 1) w ( S k ), then w ( S k ′ ) = w ( S k ) − w ( C ′ 1 ) ≤ (1 − f ( k , a + 1)) w ( S k ). By Lemma 3.1, w e h av e w ( C 1 ) ≤ f ( k , h 1 ) · w ( S k ) and it follo ws f rom F act 3.3 that W ≤ ( f ( k , h 1 ) + q X i =2 f ( k − a, h i )(1 − f ( k , a + 1)) ) · w ( S k ) ≤ q X i =1 f ( k , h i ) · w ( S k ) . Otherwise, w ( C ′ 1 ) ≤ f ( k , a + 1) · w ( S k ) and we ha v e W = w ( C ′ 1 ) + w ( C ′′ 1 ) + q X i =2 f ( k − a, h i ) · w ( S k ′ ) . Since C ′′ 1 is a minim um ( h 1 − a )-wa y s plit of G − C ′ 1 , we h a ve w ( C ′′ 1 ) ≤ f ( k − a, h 1 − a ) · w ( S k ′ ) by Lemma 3.1. It follo w s that W ≤ w ( C ′ 1 ) + f ( k − a, h 1 − a ) · w ( S k ′ ) + q X i =2 f ( k − a, h i ) · w ( S k ′ ) = w ( C ′ 1 ) + D · w ( S k ′ ) for D = f ( k − a, h 1 − a ) + P q i =2 f ( k − a, h i ) as defined in (3). Define x = w ( C ′ 1 ) /w ( S k ) and w e h a ve W ≤ ( x + (1 − x ) D ) w ( S k ). Since 0 ≤ x ≤ f ( k , a + 1), the maximum v alue of x + (1 − x ) D o v er the inte rv al [0 , f ( k , a + 1)] must b e at either x = 0 or x = f ( k , a + 1) as it is a linear function in x . This means W w ( S k ) ≤ max { D , f ( k , a + 1) + (1 − f ( k , a + 1)) D } . Therefore by Lemma 3.4, we ha v e W ≤ ( q X i =1 f ( k , h i )) · w ( S k ) . This completes the inductive step and therefore pr ov es the theorem. W e can obtain Theorem 2.1 for Algorithm iterative-split from Theorem 3.5 as follo ws (n ote that P q i =1 ( h i − 1) = k − 1): q X i =1 f ( k , h i ) = q X i =1 (2 − h i k ) h i − 1 k − 1 = 2 k − 1 q X i =1 ( h i − 1) − 1 k ( k − 1) q X i =1 h i ( h i − 1) = 2 − P q i =1  h i 2   k 2  . Appro ximation r atio for k -w a y cuts 8 F or Algorithm iterative- h -split , we can easily derive Corollary 2.2 from Theo- rem 2.1. Remark The b ound in Theorem 3.5 is tight for k − 1 = P q i =1 ( h i − 1) and therefore the appro ximation ratios in T heorem 2 .1 and Corollary 2.2 are tig ht. T o see this, consider the f ollo wing graph G that consists of the disjoint un ion of q + 1 copies H 1 , H 2 , · · · , H q , K of the complete graph K k . F or eac h H i , fi x a subset V i of h i − 1 v ertices and let E i denote edges in H i that are co v ered b y V i . Eac h edge in E i has w eigh t 1, and eac h of the r emainin g edges of H i has w eigh t ∞ . Set the weigh t of eve ry edge in K to 1. A minimum k -wa y split C k of G consists of all edges in K , bu t iterativ e- split ma y return S q i =1 E i as a k -wa y split C ′ k of G . S ince w ( C k ) =  k 2  and w ( C ′ k ) = P q i =1 | E i | = f ( k , h i )  k 2  , we ha ve w ( C ′ k ) /w ( C k ) = P q i =1 f ( k , h i ). 4 Pro of of Lemma 3.4 In th is section, we complete our p erformance analysis by pro ving Lemma 3.4: F ≤ P q i =1 f ( k , h i ), w here F = max { D , W ′ } for D = f ( k − a, h 1 − a ) + P q i =2 f ( k − a, h i ) and W ′ = f ( k , a + 1) + (1 − f ( k , a + 1)) D . F or this purp ose, we fir st derive some useful pr op erties of f ( k , h ). F act 4.1 F or al l h 1 , h 2 ≥ 0 and k ≥ m ax { h 1 + h 2 + 1 , 2 } , f ( k , h 1 + h 2 + 1) = f ( k , h 1 + 1) + f ( k − h 1 , h 2 + 1)(1 − f ( k , h 1 + 1)) . Pro of. Let e ( k , h ) denote the num b er of edges co vered by h v ertices in the complete graph K k , and m k the n umb er of edges in K k . Th en e ( k , h 1 + h 2 ) = e ( k , h 1 ) + e ( k − h 1 , h 2 ) , and thus e ( k , h 1 + h 2 ) m k = e ( k , h 1 ) m k + e ( k − h 1 , h 2 ) m k − h 1 · m k − h 1 m k . Since m k − h 1 = m k − e ( k , h 1 ), we obtain e ( k , h 1 + h 2 ) m k = e ( k , h 1 ) m k + e ( k − h 1 , h 2 ) m k − h 1 · (1 − e ( k , h 1 ) m k ) , and the lemma follo ws fr om the fact that f ( k , h ) = e ( k , h − 1) /m k . F act 4.2 F or al l a ≥ 0 , h 2 ≥ h 1 ≥ 2 , k ≥ a + h 2 , f ( k − a, h 2 ) − f ( k , h 2 ) ≤ h 2 − 1 h 1 − 1 [ f ( k − a, h 1 ) − f ( k , h 1 )] . Appro ximation r atio for k -w a y cuts 9 Pro of. ⇔ f ( k − a, h 2 ) − h 2 − 1 h 1 − 1 f ( k − a, h 1 ) ≤ f ( k , h 2 ) − h 2 − 1 h 1 − 1 f ( k , h 1 ) ⇔ − ( h 2 − h 1 )( h 2 − 1) ( k − a )( k − a − 1) ≤ − ( h 2 − h 1 )( h 2 − 1) k ( k − 1) ⇔ ( k − a )( k − a − 1) ≤ k ( k − 1) . F act 4.3 F or al l a ≥ 0 , h ≥ 2 , k ≥ a + h, f ( k − a, h − a ) + k − h h − 1 f ( k − a, h ) ≤ k − 1 h − 1 f ( k , h ) . Pro of. ⇔ a 2 + a (1 + 2 h − 4 k ) − ( h − 2 k )( k − 1) ( k − a )( k − a − 1) ≤ 2 k − h k ⇔ k ( a 2 + a (1 + 2 h − 4 k ) − ( h − 2 k )( k − 1)) ≤ (2 k − h )( k − a )( k − a − 1) ⇔ a ( a + 1)( h − k ) ≤ 0 . F act 4.4 F or al l 2 ≤ h 1 ≤ h i ( i = 2 , 3 , · · · , q ) , 0 ≤ a < h 1 , P q i =1 ( h i − 1) ≤ k − 1 , f ( k − a, h 1 − a ) + q X i =2 f ( k − a, h i ) ≤ f ( k , h 1 ) + q X i =2 f ( k , h i ) . Pro of. Let ∆ = f ( k − a, h 1 − a ) + q P i =2 f ( k − a, h i ) − f ( k , h 1 ) − q P i =2 f ( k , h i ). By F act 4.2, we hav e q X i =2 ( f ( k − a, h i ) − f ( k , h i )) ≤ q X i =2 h i − 1 h 1 − 1 ( f ( k − a, h 1 ) − f ( k , h 1 )) = k − h 1 h 1 − 1 ( f ( k − a, h 1 ) − f ( k , h 1 )) . Therefore ∆ ≤ f ( k − a, h 1 − a ) − f ( k , h 1 ) + k − h 1 h 1 − 1 ( f ( k − a, h 1 ) − f ( k , h 1 )) = f ( k − a, h 1 − a ) + k − h 1 h 1 − 1 f ( k − a, h 1 ) − k − 1 h 1 − 1 f ( k , h 1 ) . It follo w s fr om F act 4.3 that ∆ ≤ 0, whic h p ro ves the lemma. Appro ximation r atio for k -w a y cuts 10 No w, we are r eady to pr o ve Lemma 3 .4: F ≤ P q i =1 f ( k , h i ). Recall that F = max { D , W ′ } for D = f ( k − a, h 1 − a ) + P q i =2 f ( k − a, h i ) and W ′ = f ( k , a + 1) + (1 − f ( k , a + 1)) D . As D ≤ P q i =1 f ( k , h i ) by F act 4 .4, we need only sh o w that W ′ ≤ P q i =1 f ( k , h i ). T his can b e d one b y using F act 4.1 and F act 3.3 as follo w s : W ′ = f ( k , a + 1) − f ( k − a, h 1 − a ) f ( k , a + 1) + f ( k − a, h 1 − a ) + q X i =2 f ( k − a, h i )(1 − f ( k , a + 1)) = f ( k , h 1 ) + q X i =2 f ( k − a, h i )(1 − f ( k , a + 1)) (b y F act 4.1) ≤ q X i =1 f ( k , h i ) . (b y F act 3.3) 5 Concluding remarks In this pap er, w e h a ve determined the exact appro ximation ratio of a ge neral splitting algorithm iterative-split for the minimum k -wa y cut pr oblem. Th e answ er is a su r prisingly s im p le expression 2 − P q i =1  h i 2  /  k 2  , yet it tak es a some- what sub tle and in vo lv ed ind uctiv e argumen t to prov e the result. It wo uld b e in teresting to find a direct and simpler pr oof. W e n ote th at for iterative-split , the requirement that h 1 ≤ h 2 ≤ · · · ≤ h q is crucial for obtaining the ap p ro ximation ratio of the algorithm, wh ich is unkno wn if we drop the requirement. W e also n ote that if we restrict h q to b e at most h , then iterativ e- h -split , a sp ecial case of ite rativ e-split , ac hiev es the b est appro ximation ratio among all p ossib le c hoices of h 1 ≤ h 2 ≤ · · · ≤ h q . Finally , w e ma y u s e iterativ e- split as a g eneral framework for designing appro ximation algorithms for v arious cu t and partition pr oblems, and the ideas in this pap er ma y shed ligh t on the analysis of this general approac h for these problems. References [1] Goldschmidt, O., Ho c hbaum, D.: A p olynomial algorithm for the k -cut problem for fixed k . Ma thematics of Op erations Researc h 19 (1) (1994) 24– 37 A pr eliminary v ersion app eared in FOCS1988 . [2] Kamidoi, Y., Y oshida, N., Nagamochi, H.: A deterministic algorithm for finding all minimum k -wa y cuts. SIAM J ournal on Computing 36 (5) (2006) 1329– 1341 Appro ximation r atio for k -w a y cuts 11 [3] Xiao, M.: An impro ved divid e-and-conqu er algorithm for find ing all mini- m um k -w a y cuts. In : Pro ceedings of the 19th Inte rnational Sym p osium on Algorithms and C omp utation (ISAAC 2008) . (2008) 208–219 [4] Karger, D.R., S tein, C.: A new approac h to the minim um cut problem. Journal of the A CM 43 (4) (1996) 601–640 Preliminary p ortions app eared in SODA 1993 and STOC1993. [5] Th orup, M.: Minimum k-w ay cuts via d etermin istic greedy tree pac king. In: Pro ceedings of the 40th Annual A CM S ymp osium on Theory of Computing (STOC 2008). (2008) 159–1 66 [6] Do wney , R.G., Estivill-Castro, V., F ello ws, M.R., Pr ieto, E., Rosamond, F.A.: Cutting up is h ard to d o: the parameterized complexity of k-cut and related problems. Elect r. Notes Th eor. Compu t. S ci. 78 (2003) 1–14 [7] Nagamo c hi, H., Ibaraki, T.: C omputing edge connectivit y in multigraphs and capacitated graphs. S IAM Jour nal on Discrete Mathematics 5 (1) (1992) 54–66 [8] Hao, J., Or lin , J .B.: A faste r algorithm for finding the minimum cut in a graph. In: P ro ceedings of the third annual ACM-SIAM symp osium on Dis- crete algorithms (SOD A1992), P h iladelphia, P A, USA, S ociet y for Ind ustrial and Applied Mathematics (1992) 165–174 [9] Bur let, M ., Goldsc hmidt, O.: A new and imp r o ved algorithm for the 3-cut problem. Oper ations Researc h Letters 21 (5) (1997) 225–22 7 [10] Nagamo c h i, H., Ibaraki, T .: A fast algorithm for computing m inim um 3-w a y and 4-w a y cuts. Mathematical Pr ogramming 88 (3) (2000) 507–520 [11] Nagamo c h i, H., Kata y ama, S ., Ib araki, T.: A faster algorithm for computing minim um 5-w a y and 6-w a y cuts in graphs. In: Asano, T., Imai, H., Lee, D.T., Nak ano, S .-i., T okuy ama, T. (eds.) CO COON 1999. LNCS, vo l. 1627, Springer, Heidelb erg. (1999) [12] Levine, M.S.: F ast randomized alg orithms for computing minim um { 3,4,5 ,6 } -w a y c uts. In: P ro ceedings of the 11th annual A CM-SIAM symp o- sium on Discrete algorithms (SODA2 000), Philadelphia, P A, USA, So ciet y for Indu strial and Applied Mathematics (2000) 735–742 [13] Saran, H., V azirani, V.V.: Finding k-cuts within t wice the optimal. SIAM J. Comput. 24 (1) (1995 ) 101–108 A preliminary version app eared in F OCS1991. [14] Naor, J., Rab an i, Y.: T ree p ac king and approxi mating k-cuts. In : Pro ceed- ings of the t we lfth ann ual A CM-SIAM s ymp osium on d iscrete algorithms (SOD A2001), Ph iladelph ia, P A, US A, Societ y for Industr ial and Applied Mathematics (2001) 26–27 Appro ximation r atio for k -w a y cuts 12 [15] Ra vi, R., Sin ha, A.: Approxima ting k-cuts via net wo rk strength. In : Pro- ceedings of the thirteen th ann ual ACM-SIAM symp osium on Di screte al- gorithms (SODA2 002), Philadelphia, P A, USA, S ociet y for I ndustrial and Applied Mathematics (2002) 621–6 22 [16] Kap o or, S.: On m inim um 3-cuts and appr o ximating k-cuts u sing cut trees. In: Pr oceedings of the 5th International I PCO Conference on Inte ger Pro- gramming and C ombinatorial Optimization, London, UK, Sp ringer-V erlag (1996 ) 132–146 [17] Zhao, L ., Nagamo c hi, H., Ibaraki, T .: Appr o ximating th e m inim um k-w a y cut in a graph via min imum 3-wa y cuts. J. C om b . Optim. 5 (4) (2001) 397– 410 A preliminary version app eared in ISAA C1999. [18] Zhao, L., Nagamo chi, H., Ibaraki, T.: Greedy splitting algorithms for ap- pro ximating m ultiw a y partition p roblems. Math. Program. 102 (1) (2005) 167–1 83 [19] Zhao, L ., Nagamochi, H., Ibaraki, T.: On generalize d greedy splitting al- gorithms for multiw a y partition pr oblems. Discrete Applied Mathematics 143 (1-3 ) (2004) 130–14 3